DBI Analysis of Open String Bound States on Non-compact D-branes

More documents

Recommendations

Info

CHAPTER 4. CONFORMAL INVARIANCE 58in which 8β g µν = α ′ (R µν + 2∇ µ ∇ ν Φ − 1 4 H µκσH κσν)+ O ( α ′2) , (4.20)(βµν B = α ′ − 1 )2 ∇κ H κµν + ∇ κ ΦH κµν + O ( α ′2) , (4.21)( D − 26β Φ = α ′ 6α ′ − 1 2 ∇2 Φ + ∇ κ Φ∇ κ Φ − 1 )24 H κµνH κµν + O ( α ′2) , (4.22)where we usedH κµν = ∂ κ B µν + ∂ µ B νκ + ∂ ν B κµ .These are the generalized Einstein equations for the fields under consideration. Notethat the expansion in α ′ is only valid if the typical length scale <strong>of</strong> space-time (usuallyexpressed in terms <strong>of</strong> the Ricci scalar) is much larger than α ′ , or equivalently if spacetimeis only slightly curved at the string length scale (recall that l 2 s = 2α ′ ). The actionfor the scalar field, being one order higher in α ′ , can be considered as a first orderperturbative correction.Demanding that Weyl invariance is still respected implies that all above mentionedβ-functions should vanish, in turn demanding that the world sheet theory is describedby a conformal field theory, i.e. a scale invariant theory. It should be noted that whenβ g = β B = 0, we are left withT α α = − 1 2 βΦ R (2) , (4.23)which as we saw is exactly the conformal anomaly that arises when one quantizes atwo-dimensional conformal field theory. Hence, when β g = β B = 0, Eq. 4.18 describesa conformal field theory with central charge equaling β Φ . 9 Since β Φ should also vanish,we are left with an anomaly free conformal field theory.Furthermore, it is worth pointing out that when all above β-functions are set to zero,what remains are second order differential equations for the background fields. As itturns out, these can be regrouped into one single action,S = 12κ 2 0∫d D X √ −Ge −2φ [R + 4∇ µ Φ∇ µ Φ − 1 12 H µνλH µνλ2 (D − 26)−3α ′ + O ( α ′)] , (4.24)in which κ 0 represents a constant with no physical significance, as it can be absorbed ina redefinition <strong>of</strong> Φ. Varying this action with respect to the background fields results insaid differential equations. Remark that from this, if one splits Φ into its expectationvalue Φ 0 and the deviation there<strong>of</strong>, Φ −→ Φ 0 + Φ, one can identify g s = e Φ 0. In otherwords, the expectation value <strong>of</strong> the scalar field Φ defines the string coupling constant.If so wanted, one could redefine the metric G µν −→ ˜G µν by multiplying with a suitable8 Readers curious as to the details <strong>of</strong> the derivations <strong>of</strong> these β-functions are referred to §2.1.3 in [5].9 It can be shown that when β g = β B = 0, β Φ is constant.
CHAPTER 4. CONFORMAL INVARIANCE 59power <strong>of</strong> e Φ in order to bring the gravitational part <strong>of</strong> the action to the standard Hilbertform 1 ∫2κ d X√D − ˜GR, allowing one to identifyκ ≡ κ 0 e Φ 0= √ 8πG N , (4.25)with G N Newton’s gravitational constant. Details can be found in §3.7 in [18] or §2.7in [12].4.3 SummaryThe few preceding pages attempted to highlight some <strong>of</strong> the essential characteristics <strong>of</strong>a conformal field theory. Then a short review was given <strong>of</strong> the problems associated withthe generalization <strong>of</strong> (bosonic) string theory from flat space-time to curved space-times,further including the Kalb-Ramond and scalar Φ background fields, and it was shownthat the resolution <strong>of</strong> these problems demanded that the world sheet theory be describedby a conformal field theory. It was further brought forward that the expectation value<strong>of</strong> the Φ field in fact determines the string coupling g s .
Page 1:
Master ThesisDBI <
Page 5 and 6:
ContentsAcknowledgements 1Samenvatt
Page 7:
7.1.4 Perturbed equations o
Page 10 and 11:
SamenvattingDe zogehete snaartheori
Page 12 and 13:
Chapter 1Introduction“Smokey, my
Page 14:
CHAPTER 1. INTRODUCTION 6sight seem
Page 20 and 21: Chapter 2Bosonic String</st
Page 22 and 23: CHAPTER 2. BOSONIC STRINGS 14moment
Page 24 and 25: CHAPTER 2. BOSONIC STRINGS 162.3 Eq
Page 26 and 27: CHAPTER 2. BOSONIC STRINGS 18we see
Page 28 and 29: CHAPTER 2. BOSONIC STRINGS 20which
Page 30 and 31: CHAPTER 2. BOSONIC STRINGS 22which
Page 32 and 33: CHAPTER 2. BOSONIC STRINGS 24from w
Page 34 and 35: CHAPTER 2. BOSONIC STRINGS 26to use
Page 36 and 37: CHAPTER 2. BOSONIC STRINGS 28This a
Page 38 and 39: CHAPTER 2. BOSONIC STRINGS 30World
Page 40 and 41: CHAPTER 2. BOSONIC STRINGS 32This g
Page 42 and 43: CHAPTER 2. BOSONIC STRINGS 34<stron
Page 44 and 45: CHAPTER 2. BOSONIC STRINGS 36For th
Page 46 and 47: Chapter 3Superstrings“One <strong
Page 48 and 49: CHAPTER 3. SUPERSTRINGS 40In D dime
Page 50 and 51: CHAPTER 3. SUPERSTRINGS 423.3.1 Clo
Page 52 and 53: CHAPTER 3. SUPERSTRINGS 443.4 Quant
Page 54 and 55: CHAPTER 3. SUPERSTRINGS 46freedom t
Page 56 and 57: CHAPTER 3. SUPERSTRINGS 48hence, fe
Page 58 and 59: CHAPTER 3. SUPERSTRINGS 50miraculou
Page 60 and 61: CHAPTER 3. SUPERSTRINGS 52associate
Page 62 and 63: CHAPTER 4. CONFORMAL INVARIANCE 54t
Page 64 and 65: CHAPTER 4. CONFORMAL INVARIANCE 56w
Page 68 and 69: Chapter 5Branes“Many speak <stron
Page 70 and 71: CHAPTER 5. BRANES 62but if the stri
Page 72 and 73: CHAPTER 5. BRANES 64meaning that th
Page 74 and 75: CHAPTER 5. BRANES 66i.e. our origin
Page 76 and 77: CHAPTER 5. BRANES 68The presence <s
Page 78 and 79: CHAPTER 5. BRANES 70heuristic argum
Page 80 and 81: CHAPTER 5. BRANES 72where F 12 is t
Page 83 and 84: Part IIIResearch75
Page 85 and 86: CHAPTER 6. SETTING 77it does. This
Page 87 and 88: CHAPTER 6. SETTING 79of</st
Page 89 and 90: CHAPTER 6. SETTING 81It was mention
Page 91 and 92: CHAPTER 6. SETTING 83through the in
Page 93 and 94: Chapter 7A First Example“Le temps
Page 95 and 96: CHAPTER 7. A FIRST EXAMPLE 877.1.3
Page 99 and 100: CHAPTER 7. A FIRST EXAMPLE 91with
Page 103 and 104: CHAPTER 7. A FIRST EXAMPLE 95This t
Page 105 and 106: CHAPTER 8. COMPUTATIONS 97and the i
Page 107 and 108: CHAPTER 8. COMPUTATIONS 998.4 Regul
Page 109 and 110: CHAPTER 8. COMPUTATIONS 1018.5 Equa
Page 111 and 112: CHAPTER 8. COMPUTATIONS 103which de
Page 113: Chapter 9Conclusion & Final Thought
Page 116 and 117:
BIBLIOGRAPHY 108[16] Polchinski J.,
Page 119:
⌈I’m so glad you cameI’m so g
show all

DBI Analysis of Open String Bound States on Non-compact D-branes

Create successful ePaper yourself

Delete template?

Save as template?